The youngster won 6-2, 7-6 (3) to secure his third ATP Tour title and he was delighted with the victory.
"I'm always really happy to play here. I always enjoy my time here and to finally win the title is something special for me," he said.
"It's an amazing start to the season for me and I hope I can keep this level."
Frenchman opponent Moutet failed to get the job done, but he can be proud of his performances throughout the competition, with victories over top seed Stan Wawrinka and No. 4 seed Milos Raonic.
"Unfortunately, I couldn't win tonight because the opponent was too good. But I will remember this all my life," he said.
Both players will be rewarded with career-high FedEx Rankings on Monday, with Rublev expected to make his Top 20 debut and Moutet hitting the Top 75 for the first time.
Rublev is the third youngest champion in tournament history, only behind Andy Murray who had won here in 2008 and 2009.